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Mirrors > Home > MPE Home > Th. List > elrnmpt2 | Structured version Visualization version Unicode version |
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rngop.1 | |
elrnmpt2.1 |
Ref | Expression |
---|---|
elrnmpt2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 | |
2 | 1 | rnmpt2 6770 | . . 3 |
3 | 2 | eleq2i 2693 | . 2 |
4 | elrnmpt2.1 | . . . . . 6 | |
5 | eleq1 2689 | . . . . . 6 | |
6 | 4, 5 | mpbiri 248 | . . . . 5 |
7 | 6 | rexlimivw 3029 | . . . 4 |
8 | 7 | rexlimivw 3029 | . . 3 |
9 | eqeq1 2626 | . . . 4 | |
10 | 9 | 2rexbidv 3057 | . . 3 |
11 | 8, 10 | elab3 3358 | . 2 |
12 | 3, 11 | bitri 264 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wceq 1483 wcel 1990 cab 2608 wrex 2913 cvv 3200 crn 5115 cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: qexALT 11803 lsmelvalx 18055 efgtlen 18139 frgpnabllem1 18276 fmucndlem 22095 mbfimaopnlem 23422 tglnunirn 25443 tpr2rico 29958 mbfmco2 30327 br2base 30331 dya2icobrsiga 30338 dya2iocnrect 30343 dya2iocucvr 30346 sxbrsigalem2 30348 cntotbnd 33595 eldiophb 37320 elicores 39760 volicorescl 40767 |
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